   Chapter 11.6, Problem 23E

Chapter
Section
Textbook Problem

Use the Ratio Test to determine whether the series is convergent or divergent. ∑ n = 1 ∞ 2 ⋅ 4 ⋅ 6 ⋅ ⋅ ⋅ ⋅ ⋅ ( 2 n ) n !

To determine

Whether the series is convergent or divergent by using the ratio test.

Explanation

1) Concept:

Use the ratio test

2) The ratio test:

(i) If limnan+1an=L<1, then the series n=1an is absolutely convergent. (and therefore convergent).

(ii) If limnan+1an=L>1 or limnan+1an= then the series n=1an is divergent.

(iii) If limnan+1an=1, then the ratio test is inconclusive, that is, no conclusion can be drawn about the convergence or divergence of an.

3) Given:

n=12·4·6· · ·(2n)n!

4) Calculation:

Consider,n=1an=n=12·4·6· · ·(2n)n!

Here, an=2·4·6· · ·(2n)n! and an+1=2·4·6· · ·(2n+2)(n+1)!

By the ratio test, consider,

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